Difference-Quadrature Schemes for Nonlinear Degenerate Parabolic Integro-PDE

DIFFERENCE-QUADRATURE SCHEMES FOR NONLINEARDEGENERATE PARABOLIC INTEGRO-PDE

I. H. BISWAS, E. R. JAKOBSEN, AND K. H. KARLSEN

Abstract. We derive and analyze monotone difference-quadrature schemesfor Bellman equations of controlled Levy (jump-diffusion) processes. These

equations are fully non-linear, degenerate parabolic integro-PDEs interpreted

in the sense of viscosity solutions. We propose new “direct” discretizations ofthe non-local part of the equation that give rise to monotone schemes capable

of handling singular Levy measures. Furthermore, we develop a new general

theory for deriving error estimates for approximate solutions of integro-PDEs,which thereafter is applied to the proposed difference-quadrature schemes.

Contents

1. Introduction 12. Well-posedness & regularity results for the Bellman equation 43. Difference-Quadrature schemes for the Bellman equation 54. Error estimates for general monotone approximations 85. New approximations of the non-local term 105.1. Finite Levy measures 115.2. Unbounded Levy measures I 115.3. Unbounded Levy measures II 146. Error estimates for a switching system approximation 177. The Proof of Theorem 4.2 19Appendix A. An example of a monotone discretization of Lα 22References 22

1. Introduction

In this article we derive and analyze numerical schemes for fully non-linear,degenerate parabolic integro partial differential equations (IPDEs) of Bellman type.To be precise, we consider the initial value problem

ut + supα∈A

{− Lα[u](t, x) + cα(t, x)u− fα(t, x)− Jα[u](t, x)

}= 0 in QT , (1.1)

u(0, x) = g(x) in RN , (1.2)

Date: June 8, 2009.

2000 Mathematics Subject Classification. Primary 45K05, 65M12; 49L25,65L70.Key words and phrases. Integro-partial differential equation, viscosity solution, finite difference

scheme, error estimate, stochastic optimal control, Levy process, Bellman equation.This work was supported by the Research Council of Norway (NFR) through the project

”Integro-PDEs: Numerical methods, Analysis, and Applications to Finance”. The work of K. H.

Karlsen was also supported trough a NFR Outstanding young Investigator Award. This articlewas written as part of the the international research program on Nonlinear Partial Differential

Equations at the Centre for Advanced Study at the Norwegian Academy of Science and Lettersin Oslo during the academic year 2008–09.

1

2 I. H. BISWAS, E. R. JAKOBSEN, AND K. H. KARLSEN

where QT := (0, T ]× RN and

Lα[φ](t, x) := tr[aα(t, x)D2φ

]+ bα(t, x)Dφ,

Jα[φ](t, x) :=∫

RM\{0}

(φ(t, x+ ηα(t, x, z))− φ− 1|z|≤1η

α(t, x, z)Dφ)ν(dz),

for smooth bounded functions φ. Equation (1.1) is convex and non-local. Thecoefficients aα, ηα, bα, cα, fα, g are given functions taking values respectively in SN(N × N symmetric matrices), RN , RN , R, R, and R. The Levy measure ν(dz) isa positive, possibly singular, Radon measure on RM\{0}; precise assumptions willbe given later.

The non-local operators Jα can be pseudo-differential operators. Specifyingη ≡ z and ν(dz) = K

|z|N+γ dz, γ ∈ (0, 2), give rise to the fractional Laplace operatorJ = (−∆)γ/2. These operators are allowed to degenerate since we allow η = 0for z 6= 0. The second order differential operator Lα is also allowed to degeneratesince we only assume that the diffusion matrix aα is nonnegative definite. Due tothese two types of degeneracies, equation (1.1) is degenerate parabolic and there is no(global) smoothing of solutions in this problem (neither “Laplacian” nor “fractionalLaplacian” smoothing). Therefore equation (1.1) will have no classical solutions ingeneral. From the type non-linearity and degeneracy present in (1.1) the naturaltype of weak solutions are the viscosity solutions [20, 25]. For a precise definitionof viscosity solution of (1.1) we refer to [27]. In this paper we will work withHolder/Lipschitz continuous viscosity solution of (1.1)-(1.2). For other works onviscosity solutions and IPDEs of second order, we refer to [3, 4, 5, 7, 6, 10, 15, 27,28, 37, 40] and references therein.

Nonlocal equations such as (1.1) appear as the dynamic programming equationassociated with optimal control of jump-diffusion processes over a finite time horizon(see [37, 39, 12]). Examples of such control problems include various portfoliooptimization problems in mathematical finance where the risky assets are drivenby Levy processes. The linear pricing equations for European and Asian options inLevy markets are also of the form (1.1) if we take A to be a singleton. For moreinformation on pricing theory and its relation to IPDEs we refer to [18].

For most nonlinear problems like (1.1)-(1.2), solutions must be computed by anumerical scheme. The construction and analysis of numerical schemes for nonlinearIPDEs is a relatively new area of research. Compared to the PDE case, there arecurrently only a few works available. Moreover, it is difficult to prove that suchschemes converge to the correct (viscosity) solution. In the literature there are twomain strategies for the discretization the non-local term in (1.1). One is indirectin the sense that the Levy measure is first truncated to obtain a finite measureand then the corresponding finite integral term is approximated by a quadraturerule. Regarding this strategy, we refer to [18, 19] (linear or obstacle problems)and [29, 16] (general non-linear problems). The other approach is to discretize theintegral term directly. Now there are 3 different cases to consider depending onwhether (i)

∫|z|<1

ν(dz) <∞, (ii)∫|z|<1

|z|ν(dz) <∞, or (iii)∫|z|<1

|z|2ν(dz) <∞.Case (i) is the simplest one and has been considered by many authors, see, e.g.,[33, 18, 14, 22, 2, 29] and references therein. Case (ii) was considered in [1, 35, 24],and case (iii) in [35, 24]. Most of the cited papers restrict their attention to linear,non-degenerate, one-dimensional equations or obstacle problems for such equations.

One of the contributions of this paper is a class of direct approximations of thenon-local part of (1.1), giving rise to new monotone schemes that are capable of han-dling singular Levy measures and moreover are supported by a theoretical analysis.The proposed schemes are new also in the linear case. As in [1] (cf. also [34] for arelated approach), the underlying idea is to perform integration by parts to obtain a

DIFFERENCE-QUADRATURE SCHEMES FOR IPDES 3

bounded “Levy” measure and an integrand involving derivatives of the unknown so-lution. In [1], one-dimensional, constant coefficients, linear equations (and obstacleproblems) are discretized under the assumption

∫|z|<1

|z| ν(dz) <∞. Their schemesare high-order and non-monotone, but not supported by rigorous stability and con-vergence results. In this paper we discretize general non-linear, multi-dimensional,non-local equations without any additional restrictive integrability condition on theLevy measure. More precisely, we provide monotone difference-quadrature schemesfor (1.1)-(1.2) and prove under weak assumptions that these schemes converge witha rate to the exact viscosity solution of the underlying IPDE. The schemes we putforward and our convergence results apply in much more general situations thanthose previously treated in the literature.

The second main contribution of this paper is a theory of error estimates for aclass of monotone approximations schemes for the initial value problem (1.1)-(1.2).We use this theory to derive error estimates for the proposed numerical schemes. ForIPDEs in general and non-linear IPDEs in particular, there are few error estimatesavailable, see [19, 36] for linear equations and [29, 11, 16] for non-linear equations.

Error estimates involving viscosity solutions first appeared in 1984 for first orderPDEs [21], in 1997/2000 for convex 2nd order PDEs [30, 31], and in 2005/2008 forIPDEs [19, 29]. The results obtained for IPDEs, including those in this paper, areextensions of the results known for convex second order PDEs, which are based onKrylov’s method of shaking the coefficients [31]. Krylov’s method produces smoothapproximate subsolutions of the equation (or scheme) that, via classical comparisonand consistency arguments, imply one-sided error estimates. Based on this idea,there are currently two types of error estimates for convex second order PDEs: (i)optimal rates applying to specific schemes and equations (cf., e.g., [32, 26]) and(ii) sub-optimal rates that apply to “any” monotone consistent approximation (cf.,e.g., [8, 31]). In particular, type (i) results apply when you have a priori regularityresults for the scheme, while type (ii) results do not require this.

In this paper we provide error estimates of type (ii), whereas earlier results forIPDEs are of type (i), see [19, 29, 11, 16]. The problem with type (i) results is thedifficulty in establishing the required priori regularity estimates. In the PDE casethis can be achieved for particular schemes [32, 26], and attempts to generalize theseschemes to the IPDE setting have only been partially successful [11, 16], since therequired regularity estimates have been obtained only through unnaturally strongrestrictions on the non-local terms. In [16] the Levy measure is bounded and in[11] the Levy measure is either bounded or the integral term is independent of xwith an (essentially) one-dimensional Levy measure. Of course, by a truncationprocedure only bounded Levy measures need to be considered [18, 29], but suchapproximations may not be accurate and the resulting error estimates blow up asthe truncation parameter tends to zero. An advantage of the error estimates in thepresent paper is that they apply without any such restrictions. In particular, wecan handle naturally any singular Levy measures directly in our framework.

To prove our results we extend the approach of [8] to the non-local setting. Tothis end, we have to invoke a switching system approximation of (1.1) (see Section6). Switching systems of this generality have not been studied before. In paper[12], we provide well-posedness, regularity, and continuous dependence results forsuch systems. We also prove that the value function of a combined switching andcontinuous control problem solve the switching system under consideration.

The remaining part of this paper is organized as follows: First of all, we shallend this introduction by listing some relevant notation. In Section 2 we list afew standing assumptions and provide corresponding well-posedness and regularityresults for the IPDE problem (1.1)-(1.2). In Section 3 we present a rather general


approximation scheme for this problem, and show that it is consistent, monotone,and convergent. Error estimates for general monotone approximation schemes arestated in Section 4. In Section 5 we present new direct discretizations of the non-local term in (1.1), and prove that these discretizations are consistent, monotone,and also satisfy the requirements introduced in Section 3. The switching systemapproximation of (1.1) is introduced and analyzed in Section 6. The obtainedresults are utilized in Section 7 to prove the error estimate stated Section 4. Finally,in Appendix A we give a standard example of a (monotone) discretization of thelocal PDE part of (1.1) that satisfies the requirements of Section 3.

We now introduce the notation we will use in this paper. By C,K we meanvarious constants which may change from line to line. The Euclidean norm on anyRd-type space is denoted by | · |. For any subset Q ⊂ R×RN and for any bounded,possibly vector valued, function on Q, we define the following norms,

|w|0 := sup(t,x)∈Q

|w(t, x)|, |w|1 = |w|0 + sup(t,x) 6=(s,y)

|w(t, x)− w(t, y)||t− s| 12 + |x− y|

.

Note that if w is independent of t, then |w|1 is the Lipschitz (or W 1,∞) norm of w.We use Cb(Q) to denote the space of bounded continuous real valued functions onQ. Let ρ(t, x) be a smooth and non-negative function on R × RN with unit massand support in {0 < t < 1}×{|x| < 1}. For any ε > 0, we define the mollifier ρε by

ρε(t, x) :=1

εN+2ρ( tε2,x

ε

). (1.3)

In this paper we denote by h the vector

h = (∆t,∆x,∆z) > 0,

and any dependence on ∆t, ∆x, or ∆z will be denoted by subscript h. The grid isdenoted by Gh and is a subset of QT which need not be uniform or even discrete ingeneral. We also set G0

h = Gh ∩ {t = 0} and G+h = Gh ∩ {t > 0}.

2. Well-posedness & regularity results for the Bellman equation

In this section we give some relevant well-posedness and regularity results for theBellman equation (1.1)-(1.2). To this end, we impose the following assumptions:(A.1) The control set A is a separable metric space. For any α ∈ A, aα =

12σ

ασαT , and σα, bα, cα, fα, ηα are continuous in α for all x, t, z.

(A.2) There is a positive constant K such that for all α ∈ A,

|g|1 + |σα|1 + |bα|1 + |cα|1 + |fα|1 ≤ K.

(A.3) For every α ∈ A and z ∈ RM there is an Λ ≥ 0 such that

|e−Λ|z|ηα(·, ·, z)|1 ≤ K(|z| ∧ 1) and |e−Λ|·|ηα(t, x, ·)|1 ≤ K.

(A.4) ν is a positive Radon measure on RM \ {0} satisfying∫0<|z|≤1

|z|2ν(dz) +∫|z|≥1

e(Λ+ε)|z|ν(dz) ≤ K

for some K ≥ 0, ε > 0 where Λ is defined in (A.3).Sometimes we need the following stronger assumptions than (A.3) and (A.4):

(A.4’) ν is a positive Radon measure having a density k(z) satisfying

0 ≤ k(z) ≤ e−(Λ+ε)|z|

|z|M+γfor all z ∈ RM \ {0},

for some γ ∈ (0, 2), ε > 0, where Λ is defined in (A.3).


(A.5) Assume that (A.3) holds and let γ as in (A.4’). There is a constant K suchthat for every α ∈ A and z ∈ RM

|Dkzηα(·, ·, z)|0 + |Dl

xηα(·, ·, z)|0 ≤ KeΛ|z|,

for allk = l = 1 when γ = 0,k, l ∈ {1, 2} when γ ∈ (0, 1),k ∈ {1, 2, 3, 4}, l ∈ {1, 2} when γ ∈ [1, 2).

Assumptions (A.1)–(A.4) are standard and general. The assumptions on thenon-local term are motivated by applications in finance. Almost all Levy modelsin finance are covered by these assumptions. It is easy to modify the results inthis paper so that they apply to IPDEs under different assumptions on the Levymeasures, e.g., to IPDEs of fractional Laplace type where there is no exponentialdecay of the Levy measure at infinity. Finally, assumption (A.5) is not strictlyspeaking needed in this paper. We use it in some results because it simplifies someof our error estimates.

Under these assumptions the following results hold:

Proposition 2.1. Assume (A.1)–(A.4).(a) There exists a unique bounded viscosity solution u of the initial value problem(1.1)–(1.2) satisfying |u|1 <∞.(b) If u1 and u2 are respectively viscosity sub and supersolutions of (1.1) satisfyingu1(0, ·) ≤ u2(0, ·), then u1 ≤ u2.

The precise definition of viscosity solutions for the non-local problem (1.1)–(1.2)and the proof of Proposition 2.1 can be found in [27], for example.

3. Difference-Quadrature schemes for the Bellman equation

Now we explain how to discretize (1.1)–(1.2) by convergent monotone schemeson a uniform grid (for simplicity). We start by the spatial part and approximatethe non-local part Jα as explained later in Section 5 and the local PDE part Lα bya standard monotone scheme (cf. [33] and Appendix A). The result is a system ofODEs in ∆xZN × (0, T ):

ut + supα∈A

{− Lαh [u](t, x) + cα(t, x)u− fα(t, x)− Jαh [u](t, x)

}= 0,

where Lh and Jh are monotone, consistent approximations of L and J , respectively.Then we discretize in the time variable using two separate θ-methods, one for

the differential part and one for the integral part. For ϑ, θ ∈ [0, 1], the fully discretescheme reads

Unβ = Un−1β −∆t sup

α∈A

{− θLαh [U ]nβ − (1− θ)Lαh [U ]n−1

β + cα,n−1β Un−1

β (3.1)

− fα,n−1β − ϑJαh [U ]nβ − (1− ϑ)Jαh [U ]n−1

β

}in G+

h ,

U0β = g(xβ) in G0

h, (3.2)

where Gh = ∆xZN × ∆t{0, 1, 2, . . . , T∆t} and Unβ = U(tn, xβ), fα,nβ = fα(tn, xβ),etc., for tn = n∆t (n ∈ N0) and xβ = β∆x (β ∈ ZN ).

The approximations Lh and Jh are consistent, satisfying

|Lα[φ]− Lαh [φ]| ≤ KL(|D2φ|0∆x+ |D4φ|0∆x2), (3.3)

|Jα[φ]− Jαh [φ]| ≤ KI∆x

{|D2φ|0 when γ = [0, 1),(|D2φ|0 + |D4φ|0) when γ = [1, 2),

(3.4)


for smooth bounded functions φ and where γ ∈ (0, 2) is defined in (A.4’). They arealso monotone in the sense that they can be written as

Lαh [φ](tn, xβ) =∑β∈ZN

lα,nh,β,β

[φ(tn, xβ)− φ(tn, xβ)

]with lα,n

h,β,β≥ 0, (3.5)

Jαh [φ](tn, xβ) =∑β∈ZN

jα,nh,β,β

[φ(tn, xβ)− φ(tn, xβ)

]with jα,n

h,β,β≥ 0, (3.6)

for any β ∈ ZN and n ∈ N0. We also assume without loss of generality thatj·,··,β,β = 0 = l·,··,β,β for all β ∈ ZN . The sum (3.5) is always finite, while the sum(3.6) is finite if the Levy measure ν is compactly supported. With γ ∈ [0, 2) definedin (A.4’) and ∆x < 1, we also have that

lα,nβ

:=∑β∈ZN

lα,nh,β,β

≤ Kl supα

{|aα|0∆x−2 + |bα|0∆x−1

}, (3.7)

jα,nβ

:=∑β∈ZN

jα,nh,β,β

≤ Kj∆x−1. (3.8)

From (3.3) and (3.4) it immediately follows that the scheme (3.1) is a consistentapproximation of (1.1), with the truncation error bounded by

12|φtt|0∆t+ sup

α,n

{|Lα[φ]n − Lαh [φ]n|0 + |Jα[φ]n − Jαh [φ]n|0

+ (1− θ)|Lα[φ]n−1 − Lα[φ]n|0 + (1− ϑ)|Jα[φ]n−1 − Jα[φ]n|0},

(3.9)

for smooth functions φ. The last two terms are again bounded by

∆t supα

{|Lα[φt]|0 + |Jα[φt]|0

}≤ K∆t

{|∂tDφ|0 + |∂tD2φ|0

}. (3.10)

Under a CFL condition, the scheme (3.1) is also monotone, meaning that thereare numbers bm,k

β,β(α) ≥ 0 such that it can be written as

supα

{bn,nβ,β

(α)Unβ −∑β 6=β

bn,nβ,β

(α)Unβ −∑β

bn,n−1

β,β(α)Un−1

β −∆tfn−1,α

β

}= 0, (3.11)

for all (xβ , tn) ∈ G+h . From (3.5) and (3.6), we see that

bn,mβ,β

(α) =

1 + ∆tθ lα,mβ

+ ∆tϑ jα,mβ

when m = n,

1−∆t[(1− θ)lα,m

β+ (1− ϑ)jα,m

β− cα,m

β

]when m = n− 1,

bn,mβ,β

(α) =

{∆tθlα,m

h,β,β+ ∆tϑjα,m

h,β,βwhen m = n,

∆t(1− θ)lα,mh,β,β

+ ∆t(1− ϑ)jα,mh,β,β

when m = n− 1,

where β 6= β and other choices of m give zero. These coefficients are positiveprovided the following CFL condition holds:

∆t[(1− θ)lα,mβ + (1− ϑ)jα,mβ − cα,mβ

]≤ 1 for all α, β,m, (3.12)

or alternatively by (3.7) and (3.8), if aα 6≡ 0, cα ≥ 0, ∆x < 1,

∆t[(1− θ)KlC∆x−2 + (1− ϑ)Kj∆x−1

]≤ 1.

Existence, uniqueness, and convergence results for the above approximationscheme are collected in the next theorem, while error estimates are postponed toTheorem 4.3 in Section 4.


Theorem 3.1. Assume (A.1)–(A.3), (A.4’), (3.3)–(3.8), and (3.12).

(a) There exists a unique bounded solution Uh of (3.1)–(3.2).

(b) The scheme is L∞-stable, i.e. |Uh| ≤ esupα |cα|0tn

[|g|0 + tn supα |fα|0

].

(c) Uh converge uniformly to the viscosity solution u of (1.1)–(1.2) as h→ 0.

Proof. The existence and uniqueness of bounded solutions follow by an inductionargument. Consider t = tn and assume Un−1 is a given bounded function. Forε > 0 we define the operator T : Un → Un by

TUnβ = Unβ − ε · (left hand side of (3.11)) for all β ∈ ZM .

Note that the fixed point equation Un = TUn is equivalent to equation (3.1).Moreover, for sufficiently small ε, T is a contraction operator on the Banach spaceof bounded functions on ∆xZN under the sup-norm. Existence and uniquenessthen follows from the fixed point theorem (for Un) and for all of U by inductionsince U0 = g|G0

his bounded.

To see that T is a contraction we use the definition and sign of the b-coefficients:

TUnβ − TUnβ

≤ supα

{[1− ε[1 + ∆t(θlα,nβ + ϑjα,nβ )]

](Unβ − Unβ ) + ε∆t(θlα,nβ + ϑjα,nβ )|Un· − Un· |0

}≤ (1− ε)|Un· − Un· |0,

provided 1 − ε(1 + ∆t(θlα,nβ + ϑjα,nβ ))] ≥ 0 for all α, β, n. Taking the supremumover all β and interchanging the role of U and U proves that T is a contraction.

Much the same argument, utilizing (3.11), establishes that Uh is bounded by aconstant independent of h:

|Un|0 ≤ (1 + ∆t supα|cα|0)n

[|g|0 +n∆t sup

α|fα|0

]≤ esupα |c

α|0tn[|g|0 + tn sup

α|fα|0

].

In view of this bound, the convergence of Uh to the solution u of (1.1)–(1.2) followsby adapting the Barles-Souganidis argument [9] to the present non-local context.Alternatively, convergence follows from Theorem 4.3 if we also assume (A.5). �

Remark 3.1.

a. One suitable choice of Jαh will be derived in Section 5, while for Lαh thereare several choices that satisfies (3.3) and (3.5), e.g., the scheme by Bonnans andZidani [13] or the (standard) schemes of Kushner [33]. In Appendix A we show thatone of the schemes of Kushner fall into our framework if aα is diagonally dominant.

b. For the differential part, the choices θ = 0, 1, and 1/2 give explicit, implicit,and Crank-Nicholson discretizations. When ϑ > 0, the integral term is evaluatedimplicitly. This leads to linear systems with full matrices and is not used much inthe literature.

c. By parabolic regularity D2 ∼ ∂t and (3.10) is similar to ∆t|φtt|0. Whenθ = 1/2 = ϑ the scheme (3.1) (Crank-Nicholson!) is second order in time O(∆t2)and (3.9) is no longer optimal.

d. When γ = 0 the leading error term in Jh[u] (see (3.4)) comes from differenceapproximation of the term Du

∫ην. This difference approximation also give rise

the term ∆x−1 in (3.8).


4. Error estimates for general monotone approximations

In this section we present error estimates for nonlinear general monotone ap-proximation schemes for IPDEs. As a corollary we obtain an error estimate for thescheme (3.1)–(3.2) defined in Section 3. These results, which extend those in [8] tothe non-local IPDE context, can be applied to “any” Levy-type integro operator.Earlier results apply to either linear problems, specific schemes, or restricted typesof Levy operators, see [36, 19, 29, 11]. In particular, previous error estimates donot apply to the approximation scheme (3.1).

Let us write (1.1) as ut + F [u] = 0 where F [u] := F (t, x, u,Du,D2u, u(t, ·))denotes the sup part of (1.1). We write approximations of ut + F [u] = 0 as

S(h, t, x, uh(t, x), [uh]t,x) = 0 in G+h , (4.1)

uh(0, x) = gh(x) in G0h, (4.2)

where S is the approximation of (1.1) defined on the mesh Gh ⊂ QT with “mesh”parameter h = (∆t,∆x,∆z) (time, space, quadrature parameters). The solution istypified by uh and by [uh]t,x we denote a function defined at (t, x) in terms of thevalues taken by uh evaluated at points other than (t, x). Note that the grid doesnot have to be uniform or even discrete.

We assume that (4.1) satisfies the following set of (very weak) assumptions:

(S1) (Monotonicity) There exist λ, µ ≥ 0, h0 > 0 such that, if |h| ≤ h0, u ≤ vare functions in Cb(Gh) and φ(t) = eµt(a+ bt) + c for a, b, c ≥ 0, then

S(h, t, x, r + φ(t), [u+ φ]t,x) ≥ S(h, t, x, r, [v]t,x) +b

2− λc in G+

h .

(S2) (Regularity) For each h and φ ∈ Cb(Gh), the mapping

(t, x) 7→ S(h, t, x, φ(t, x), [φ]t,x

)is bounded and continuous in G+

h and the function r 7→ S(h, t, x, r, [φ]t,x) isuniformly continuous for bounded r, uniformly in t, x.

(S3) (i) (Sub-consistency) There exists a function E1(K, h, ε) such that, forany sequence {φε}ε of smooth bounded functions satisfying

|∂β0t Dβ′φε| ≤ Kε1−2β0−|β′| in QT , for any β0 ∈ N, β′ ∈ NN ,

where |β′| =∑Ni=1 β

′i, the following inequality holds in G+

h :

S(h, t, x, φε(t, x), [φε]t,x

)≤ φεt + F (t, x, φε, Dφε, D2φ, φε(t, ·)) + E1(K, h, ε).

(S3) (ii) (Super-consistency) There exists a function E2(K, h, ε) such that,for any sequence {φε}ε of smooth bounded functions satisfying

|∂β0t Dβ′φε| ≤ Kε1−2β0−|β′| in QT , for any β0 ∈ N, β′ ∈ NN ,

the following inequality holds in G+h :

S(h, t, x, φε(t, x), [φε]t,x

)≥ ∂tφε + F (t, x, φε, Dφε, D2φ, φε(t, ·))− E2(K, h, ε).

Remark 4.1. In (S3), we typically take φε = wε ∗ ρε for some sequence (wε)εof uniformly bounded and Lipschitz continuous functions, and ρε is the mollifierdefined in Section 1.

Remark 4.2. Assumption (S1) implies monotonicity in [u] (take φ = 0), andparabolicity of the scheme (4.1) (take u = v). This last point is easier to understandfrom the following more restrictive assumption:


(S1’) (Monotonicity) There exist λ ≥ 0, K > 0 such that if u ≤ v;u, v ∈ Cb(Gh)and φ : [0, T ]→ R smooth, then

S(h, t, x, r + φ(t), [u+ φ]t,x)

≥ S(h, t, x, r, [v]t,x) + φ′(t)− K∆t|φ′′(t)|0 − λφ+(t).

It is easy to see that (S1’) implies (S1), cf. [8].

The main consequence of (S1) and (S2) is the following comparison principlesatisfied by scheme (4.1) (for a proof, cf. [8]):

Lemma 4.1. Assume (S1), (S2), g1, g2 ∈ Cb(Gh), and u, v ∈ Cb(Gh) satisfy

S(h, t, x, u(t, x), [u]t,x) ≤ g1 and S(h, t, x, v(t, x), [v]t,x) ≥ g2 in G+h .

Then, for λ and µ as in (S1),

u− v ≤ eµt|(u(0, ·)− v(0, ·))+|0 + 2tet|(g1 − g2)+|0.

The following theorem is our first main result.

Theorem 4.2 (Error Estimate). Assume (A.1)–(A.4), (S1), (S2) hold, and thatthe approximation scheme (4.1)–(4.2) has a unique solution uh ∈ Cb(Gh), for eachsufficiently small h. Let u be the exact solution of (1.1)–(1.2).

a) (Upper Bound) If (S3)(i) holds, then there exists a constant C, dependingonly on µ,K in (S1) and (A.2), such that

u− uh ≤ eµt|(g − gh)+|0 + C minε>0

(ε+ E1(|u|1, h, ε)

)in Gh.

b) (Lower Bound) If (S3)(ii) holds, then there exists a constant C, dependingonly on µ,K in (S1) and (A.2), such that

u− uh ≥ −eµt|(g − gh)−|0 − C minε>0

(ε

13 + E2(|u|1, h, ε)

)in Gh.

We prove this theorem in Section 7.

Remark 4.3. Theorem 4.2 applies to all Levy type non-local operators. Notethat the lower bound is worse than the upper bound, and may not be optimal. Incertain special cases it is possible to prove better bounds, however until now suchresults could only be obtained in the non-degenerate linear case [36, 19] or undervery strong restrictions on the non-local term [11, 16]. More information on suchnon-symmetric error bounds can be found in [8].

Remark 4.4. For a finite difference-quadrature type discretization of (1.1), thetruncation error would typically look like

|φt + F (t, x, φ,Dφ,D2φ, φ(t, ·))− S(h, t, x, φ(t, x), [φ]t,x)|

≤ K∑β0

|∂β00t Dβ′0φ|0∆tkβ0 +K

∑β1

|∂β01t Dβ′1φ|0∆xkβ1 +K

∑β2

|∂β02t Dβ′2φ|0∆zkβ2 ,

where β0 = (β00 , β′0), β1 = (β0

1 , β′1), β2 = (β0

2 , β′2) are multi-indices and kβ0 , kβ2 , kβ2

are real numbers. In this case, the function E in (S3) is obtained by taking φ := φεin the above inequality:

E1 = E2 = KK∑

β0,β1,β2

[ε1−2β0

0−|β′0|∆tkβ0 + ε1−2β0

1−|β′1|∆xkβ1 + ε1−2β0

2−|β′2|∆zkβ2

].

An optimization with respect to ε yields the final convergence rate. Observe thatthe obtained rate reflects a potential lack of smoothness in the solution.

We shall now use Theorem 4.2 to prove error estimates for the finite difference-quadrature scheme (3.1).


Theorem 4.3. Assume (A.1)–(A.3), (A.4’), (A.5), (3.3)–(3.8), (3.12) hold, andthat u and Uh are the solutions respectively of (1.1)–(1.2) and (3.1)–(3.2).

There are constants KL,KJ ≥ 0, δ > 0 such that if ∆x ∈ (0, δ) and ∆t satisfiesthe CFL condition (3.12), then in Gh,

−K(∆t1/10 + ∆x1/5) ≤ u− Uh ≤ K(∆t1/4 + ∆x1/2) for γ ∈ [0, 1),

−K(∆t1/10 + ∆x1/10) ≤ u− Uh ≤ K(∆t1/4 + ∆x1/4) for γ ∈ [1, 2).

Proof. Let us write the scheme (3.1) in abstract form (4.1). To this end, set[u]t,x(s, y) = u(t + s, x + y) and divide (3.11) by ∆t to see that (3.1) takes theform (4.1) with

S(h, tn, xβ , r, [u]tn,xβ ) = supα∈A

{bn,nβ,β(α)

∆tr −

∑β 6=β

bn,nβ,β

(α)

∆t[u]tn,xβ (0, xβ − xβ)

−∑β

bn,n−1

β,β(α)

∆t[u]tn,xβ (−∆t, xβ − xβ)

}.

By its definition (3.1), monotonicity (3.11), and consistency (3.9), this schemeobviously satisfies assumptions (S1) – (S3) if the CFL condition (3.12) holds. Inparticular, from (3.9) and (3.3), (3.4), (3.10), we find that

E1(K, h, ε) = E2(K, h, ε) =

{CK(∆tε−3 + ∆xε−1 + ∆x2ε−3), γ ∈ [0, 1)CK(∆tε−3 + ∆xε−1 + (∆x2 + ∆x)ε−3), γ ∈ [1, 2).

The result then follows from Theorem 4.2 and a minimization with respect to ε. �

Remark 4.5. The error estimate is independent of γ and robust in the sense thatit applies to non-smooth solutions.

5. New approximations of the non-local term

In this section we derive direct approximations Jαh [u] of the non-local integro termJα[u] appearing in (1.1). As in [1] (cf. also [34]), the idea is to perform integrationby parts to reduce the singularity of the measure. For the full discretization of (1.1)along with convergence analysis, we refer to Section 3.

We consider 3 cases separately: (i)∫|z|<1

ν(dz) < ∞, (ii)∫|z|<1

|z|ν(dz) < ∞,and (iii)

∫|z|<1

|z|2ν(dz) < ∞. Note that in cases (i) and (ii) we can write thenon-local operator in the form

Jα[φ](t, x) = Iα[φ](t, x)− bα(x)Dφ, (5.1)

where

Iα[φ](t, x) :=∫|z|>0

(φ(t, x+ ηα(t, x, z))− φ

)ν(dz),

bα(x) :=∫

0<|z|<1

ηα(t, x, z)ν(dz),

for smooth bounded functions φ. The reason is that Iα[φ] and bα(x) are well-definedunder assumptions (A.2), (A.3), (A.4) if either (i) or (ii) holds. Furthermore,bα(x) will be bounded and x-Lipschitz. The term bαDφ will be approximated byquadrature and upwind finite differences as in Appendix A leading to a first ordermethod. We skip the standard details and focus on the non-local term Iα[φ].

To simplify the presentation a bit, we will only consider the Cartesian x-grid{xβ}β = ∆xZN , but it is possible to consider unstructured non-degenerate families


of grids. On our grid we define a positive and 2nd order interpolation operator ih,i.e., an operator satisfying

ihφ(x) =∑β∈ZN

wβ(x)φ(xβ) with wβ(x) ≥ 0, (5.2)

|EI [φ](x)| := |φ(x)− ihφ(x)| ≤ KI∆x2|D2φ|0, (5.3)

for all x ∈ RN and where wβ(x) ≥ 0 are basis functions satisfying wβ(xβ) = δβ,βand

∑β wβ ≡ 1. Linear and multi-linear interpolation satisfy these assumptions.

Note that higher order interpolation is not monotone in general.We will also need the following monotone difference operators:

δ±r,hφ(r, y) = ± 1∆x{φ(r ±∆x, y)− φ(r, y)

}, (5.4)

∆rr,kφ(r, y) =1k2

{φ(r + k, y)− 2φ(r, y) + φ(r − k, y)

}, (5.5)

for functions φ(r, y) on R× RK for some K ∈ N. For smooth φ we have

|δ±r,hφ− ∂rφ| ≤12|φrr|0∆x, |∆rr,kφ− ∂2

rφ| ≤112|∂4rφ|0|k|2.

5.1. Finite Levy measures. Assuming∫|z|<1

ν(dz) < ∞, we approximate theterm Iα[φ] defined in (5.1) by

Iαh [φ](t, x) = Qh

[(ihφ)(t, x+ ηα(t, x, z))− φ(t, x)

],

where Qh denotes a positive quadrature rule on the z-grid {zβ}β ⊂ RM withmaximal grid spacing ∆z, satisfying

Qh[φ] =∑β∈ZM

ωβφ(zβ) with ωβ ≥ 0,

|EQ[φ]| := |∫φ(z)ν(dz)−Qh[φ]| ≤ KQ∆zkQ |DkQφ|0

∫ν(dz),

for smooth bounded functions φ, where KQ ≥ 0 and kQ ∈ N. Many quadraturemethods satisfies these requirements, e.g., compound Newton-Cotes methods oforder less than 9 and Gauss methods of arbitrary order. Note that the z-grid doesnot have to be a Cartesian grid. This method is at most 2nd order accurate because

Iα[φ] = Iαh [φ] + EI [φ(·, ·+ ηα)]∫ν(dz) + EQ[φ(·, ·+ ηα)− φ],

and it is monotone by construction, satisfying (3.6) and (3.8). The O(∆x−1) termin (3.8) comes from the discretization of the bα term in (5.1).

5.2. Unbounded Levy measures I. Now we assume that∫|z|<1

|z|ν(dz) < ∞,or more precisely that (A.4’) holds with γ < 1. We consider the one-dimensionaland multi-dimensional cases separately.

5.2.1. One-dimensional case (M = 1). Now Iα[φ] in (5.1) takes the form

Iα[φ](t, x) =∫

R\{0}

[φ(t, x+ ηα(t, x, z))− φ(t, x)

]k(z)dz.

We approximate this term by

Iαh [φ](t, x)

=∞∑n=0

[δ+z,h(ihφ)(t, x+ ηα(t, x, zn))k+

h,n − δ−z,h(ihφ)(t, x+ ηα(t, x, z−n))k−h,n

],

(5.6)


where zn = n∆x, δ±z,h is defined in (5.4), the x-interpolation ih satisfies (5.2) and(5.3). Moreover,{

k+h,n :=

∫ zn+1

znk(z)dz,

k−h,n :=∫ z−nz−(n+1)

k(z)dzand k(z) :=

{∫ z−∞ k(ζ) dζ, if z < 0,∫∞zk(ζ) dζ, if z > 0.

By (A.4’) (M = 1 and γ < 1), 0 ≤∫

R k(z)dz <∞.To derive this approximation, the key idea is to perform integration by parts:

Iα[φ](t, x) =(∫ 0

−∞+∫ ∞

0

)(φ(t, x+ ηα(t, x, z))− φ(t, x)

)k(z)dz

=∫ ∞

0

∂

∂z

(φ(t, x+ ηα(t, x, z))

)k(z)dz −

∫ 0

−∞

∂

∂z

(φ(t, x+ ηα(t, x, z))

)k(z)dz,

for bounded C1 functions φ. Write Iα[φ] = Iα,+[φ] + Iα,−[φ], and use quadrature,finite differencing, and interpolation to proceed as follows:

Iα,+[φ](t, x) :=∫ ∞

0

∂z[φ(t, x+ ηα(t, x, z))

]k(z)dz

'∞∑n=0

∂z[φ(t, x+ ηα(t, x, z))

]∣∣z=zn

k+h,n

'∞∑n=0

φ(t, x+ ηα(t, x, zn + ∆x))− φ(t, x+ ηα(t, x, zn))∆x

k+h,n

'∞∑n=0

(ihφ)(t, x+ ηα(t, x, zn + ∆x))− (ihφ)(t, x+ ηα(t, x, zn))∆x

k+h,n.

In a similar way we can discretize Iα,−[φ] and (5.6) follows.The approximation just proposed is consistent since

Iα[φ](t, x) = Iαh [φ](t, x) + EQ + EFDM + EI ,

where EQ, EFDM, and EI denote respectively the error contributions from theapproximation of the integral (1st order), the difference approximation (up-winding,1st order), and the 2nd order interpolation. These terms can be estimated asfollows:

|EQ| ≤ ∆x |∂2zφ(·+ ηα)|0

∫Rk(z)dz,

|EFDM| ≤12

∆x |∂2zφ(·+ ηα)|0

∫Rk(z)dz,

|EI | ≤ 2∆x |D2xφ(·+ ηα)|0

∫Rk(z)dz.

The discretization (5.6) is also monotone satisfying (3.6) and also (3.8) (when Iαhreplaces Jαh ). To see this, note that ihφ(xβ) = φ(xβ) and that by (A.2) η(t, x, 0) = 0.Hence we can reorganize the sum defining Iα,+h and write

Iα,+h [φ](t, xβ)

= − 1∆x

k+h,0φ(t, xβ) +

1∆x

∞∑n=1

(k+h,n−1 − k

+h,n)(ihφ)(t, x+ ηα(t, xβ , zn))

=1

∆x

∞∑n=1

(k+h,n−1 − k

+h,n)

[(ihφ)(t, xβ + ηα(t, xβ , zn))− φ(t, xβ)

].


In a similar way

Iα,−h [φ](t, xβ) =1

∆x

∞∑n=1

(k−h,n−1 − k−h,n)

[(ihφ)(t, xβ + ηα(t, xβ , z−n))− φ(t, xβ)

].

Since k is increasing on (0,∞) and decreasing on (−∞, 0),

k±h,n−1 > k±h,n,

and hence by (5.2) and∑β wβ ≡ 1, (3.6) and (3.8) hold with

jα,nh,β,β

=1

∆x

∑l∈Z\{0}

wβ(xβ + ηα(tn, xβ , zl))(ksign(l)h,|l|−1 − k

sign(l)h,|l| ) ≥ 0,

jα,nβ

=1

∆x

∑l∈Z\{0}

(ksign(l)h,|l|−1 − k

sign(l)h,|l| )

∑β

wβ(xβ + ηα(tn, xβ , zl)) =k+h,0 + k−h,0

∆x,

and k±h,0 = O(∆x1−γ). The leading O(∆x−1) term in (3.8) comes from discretizingthe bα term in (5.1).

5.2.2. Multi-dimensional case (M > 1). In this case we write Iαh [φ] of (5.1) in polarcoordinates and propose the following approximation:

Iαh [φ](t, x) =∫|y|=1

∞∑n=0

δ+r,h

[ihφ(t, x+ ηα(t, x, rny))

]kh,n(y)dSy, (5.7)

where rn = n∆x, dSy is the surface measure on the unit sphere in RM , δ±z,h isdefined in (5.4), the x-interpolation ih satisfies (5.2) and (5.3). Moreover,

kh,n(y) =∫ rn+1

rn

k(r, y)dr and k(r, y) =∫ ∞r

k(sy)sM−1ds.

By assumption (A.4’) with γ ∈ (0, 1), 0 ≤∫∞

0k(r, y)dr ≤ C <∞ for all |y| = 1.

To derive this approximation we use polar coordinates and integrate by parts inthe radial direction. Let φ be a bounded C1 function, and set

Gα(t, x, z) := φ(t, x+ ηα(t, x, z))− φ(t, x).

Then

Iα[φ](t, x) =∫

RM\{0}Gα(t, x, z)k(z)dz

=∫|y|=1

[ ∫ ∞0

Gα(t, x, ry)rM−1k(ry)dr]dSy

=∫|y|=1

[ ∫ ∞0

∂

∂rGα(t, x, ry)k(r, y)dr

]dSy,

and (5.7) follows by discretizing the inner integral as in Section 5.2.1.This is a consistent first order approximation of Iα[φ] since

Iα[φ](t, x) = Iαh [φ](t, x) + EQ + EFDM + EI ,

where EQ, EFDM, EI have the same meaning as in Section 5.2.1, and these termscan be estimated as follows:

|EQ| ≤ ∆x |D2zφ(·+ ηα)|0Mk,

|EFDM| ≤12

∆x |D2zφ(·+ ηα)|0Mk,

|EI | ≤ ∆x |D2xφ(·+ ηα)|0Mk,

where Mk =∫|y|=1

∫∞0k(r, y)dr dSy.


The approximation Iαh [φ] is also monotone, and satisfies (3.6) and (3.8). Thisfollows as in Section 5.2.1, since Iαh [φ](t, xβ) can be written as

1∆x

∫|y|=1

∞∑n=1

[kh,n−1(y)− kh,n(y)

][(ihφ)(t, xβ + ηα(t, xβ , rny))− φ(t, xβ)

]dSy,

where for fixed y, kh,n(y) is a decreasing function in n since k(r, y) decreasing inr. Moreover, lα,nβ has a term like 1

∆xkh,0(y) = O(∆x−γ) plus the leading O(∆x−1)term which comes from the discretization of the bα term in (5.1).

5.3. Unbounded Levy measures II. We assume that∫|z|<1

|z|2ν(z)dx <∞, ormore precisely that (A.4’) hold with γ ∈ [1, 2). In this case the decomposition (5.1)is not valid. Again, we consider the one-dimensional and multi-dimensional casesseparately.

5.3.1. One-dimensional Levy process (M = 1). Now the nonlocal operator takesthe form

Jα[φ](t, x) =∫

R\{0}

[φ(t, x+ ηα(t, x, z))− φ(t, x)− ηα(t, x, z)Dφ

]k(z)dz,

or, after two integrations by parts (more details are given below),

Jα[φ](t, x) = Jα,+[φ](t, x) + Jα,−[φ](t, x)− bα(t, x)Dφ, (5.8)

where bα(t, x) =∫∞−∞ ∂2

zηα(t, x, z)k(z)dz,

Jα,±[φ] = ±∫ ±∞

0

∂2z

[φ(t, x+ ηα(t, x, z))

]k(z)dz, (5.9)

k(z) =

{∫ z−∞

∫ w−∞ k(r)dr dw, for z < 0∫∞

z

∫∞wk(r)dr dw, for z > 0.

By (A.4’) (M = 1, γ < 2), 0 ≤ k(z) ≤ C|z|1−γe−(Λ+ε)|z| and k is integrable.Note that bα is bounded and x-Lipschitz, and that bαDφ can be discretized

using quadratures and finite differences as in Appendix A. This leads to a firstorder monotone (upwind) approximation – we skip the standard details.

We propose the following approximation of Jα,±[φ]:

Jα,±h [φ](t, x) =∞∑n=0

∆zz,∆z

[ihφ(t, x+ ηα(t, x, zn))

]k±h,n, (5.10)

where zn = n∆x (not n∆z!), ∆zz,∆z is defined in (5.5), the x-interpolation ihsatisfies (5.2) and (5.3). Moreover,

k+h,n =

∫ zn+1

zn

k(z)dz and k−h,n =∫ z−n

z−n−1

k(z)dz.

The approximation (5.10) can be derived from (5.9) using quadrature, finite differ-encing, and interpolation.

To obtain (5.9) and (5.8), we integrate by parts twice:∫ ∞0

[φ(t, x+ ηα(t, x, z))− φ(t, x)− ηα(t, x, z)Dφ

]k(z)dz

=[[φ(t, x+ ηα(t, x, z))− φ(t, x)− ηα(t, x, z)Dφ

](−∫ ∞z

k(w)dw)]z=∞z=0

+∫ ∞

0

∂z[φ(t, x+ ηα(t, x, z))− ηα(t, x, z)Dφ

]( ∫ ∞z

k(w)dw)dz

= 0 +[∂z[φ(t, x+ ηα(t, x, z))− ηα(t, x, z)Dφ

](− k)(z)]∞

0


+∫ ∞

0

∂2z

[φ(t, x+ ηα(t, x, z))− ηα(t, x, z)Dφ

]k(z)dz

= 0 + 0 +∫ ∞

0

∂2z

[φ(t, x+ ηα(t, x, z))

]k(z)dz −Dφ

∫ ∞0

∂2zηα(t, x, z)k(z)dz.

In view of this result and similar computations for the integral on (−∞, 0), (5.8)follows. These computations are rigorous if φ(t, x + η), ∂zφ(t, x + η), ∂2

zφ(t, x + η)and η, ∂zη, ∂

2zη are z-integrable and bounded by eΛ|z| at infinity.

The approximation is consistent and has the error expansion

Jα,±[φ](t, x) = Jα,±h [φ](t, x) + E±Q + E±FDM + E±I ,

where EQ, EFDM, EI have the same meaning as in Section 5.2.1, and these termscan be estimated as follows:

|E±Q | ≤ ∆x |∂3zφ(·+ ηα)|0

∫Rk(z)dz,

|E±FDM| ≤124

∆z2 |∂4zφ(·+ ηα)|0

∫Rk(z)dz,

|E±I | ≤ 4∆x2

∆z2|D2

xφ(·+ ηα)|0∫

Rk(z)dz.

The proposed approximation is first order accurate if ∆z = ∆x1/2, it is monotonesatisfying (3.6), and (3.8) holds if ∆z = ∆x1/2. These properties follow as in Section5.2.1, since Jα,±h [φ](t, xβ) can be written as

1∆z2

k±h,0[(ihφ)(t, x+ ηα(t, xβ , z∓1))− φ(t, xβ)

]+

1∆z2

∞∑n=1

(k±h,n+1 − 2k±h,n + k±h,n−1)[(ihφ)(t, x+ ηα(t, xβ , z±n))− φ(t, xβ)

],

and, by convexity of k(z) on (0,∞) and (−∞, 0),

k±h,n+1 − 2k±h,n + k±h,n−1 ≥ 0 for n ≥ 1.

Moreover, jα,nβ equals 2∆z2 (k+

h,0 − k+h,1 + k−h,0 − k−h,1) = O(∆x2−γ/∆z2) plus a

O(∆x−1) term from the discretization of the bα-term in (5.8). When ∆z = ∆x1/2

the leading term is the O(∆x−1) term.

5.3.2. Multi-dimensional Levy process (M > 1). Writing Jα[φ] in polar coordinatesand performing two integrations by parts in the radial direction leads to

Jα[φ](t, x) = Jα[φ](t, x)− bα(t, x)Dφ, (5.11)

where bα(t, x) =∫|y|=1

∫∞0∂2r

[ηα(t, x, ry)

]k(r, y)dr dSy and

Jα[φ](t, x) =∫|y|=1

∫ ∞0

∂2r

[φ(t, x+ ηα(t, x, ry))

]k(r, y)dr dSy,

k(s) =∫ ∞s

∫ ∞w

rM−1k(ry)dr dw.

By (A.4’) (γ < 2), k(r, y) ≤ Cr1−γe−(Λ+ε)r and thus k is r-integrable uniformlyin y. Note that bα is bounded and x-Lipschitz, and that bαDφ can be discretizedusing quadrature and finite differencing as in Appendix A. This leads to a firstorder monotone (upwind) approximation – we skip the standard details.

We propose the following approximation of Jα[φ]:

Jαh [φ](t, x) =∫|y|=1

∞∑n=0

∆rr,∆z

[(ihφ)(t, x+ ηα(t, x, rny))

]kh,n(y) dSy, (5.12)


where rn = n∆x (not n∆z!), ∆rr,∆z is defined in (5.5), the x-interpolation ihsatisfies (5.2) and (5.3), and

kh,n(y) =∫ rn+1

rn

k(r, y)dz.

The approximation (5.12) follows from (5.11) by quadrature, finite differencing,and interpolation, and the derivation of (5.11) is rigorous provided the functionsφ(t, x+η), Dzφ(t, x+η), D2

zφ(t, x+η) and η,Dzη,D2zη are z-integrable and bounded

by eΛ|z| at infinity.The approximation is consistent and has the error expansion

Jα[φ](t, x) = Jαh [φ](t, x) + EQ + EFDM + EI ,

where EQ, EFDM, EI have the same meaning as in Section 5.3.1, and can beestimated as follows:

|EQ| ≤ ∆x |D3zφ(·+ ηα)|0Mk,

|EFDM| ≤124

∆z2 |D4zφ(·+ ηα)|0Mk,

|EI | ≤ 4∆x2

∆z2|D2

xφ(·+ ηα)|0Mk,

where Mk :=∫|y|=1

∫∞0k(r, y)dr dSy. Whenever ∆z = ∆x1/2, this is a first order

approximation. Moreover, the approximation is monotone satisfying (3.6) and,whenever ∆z = ∆x1/2, it also satisfies (3.8). This follows as in Section 5.2.1 sinceJαh [φ](t, xβ) can be written as an integral over {|y| = 1} with integrand

1∆z2

[(ihφ)(t, xβ + ηα(t, xβ , r−1y))− φ(t, xβ)

]kh,0(y)

+1

∆z2

∞∑n=1

(kh,n+1(y)− 2kh,n(y) + kh,n−1(y)

)×[(ihφ)(t, xβ + ηα(t, xβ , rny))− φ(t, xβ)

]].

Furthermore, for each fixed y, k(r, y) is convex on (0,∞) and thus

kh,n+1(y)− 2kh,n(y) + kh,n−1(y) ≥ 0 for n ≥ 1 and |y| = 1.

Remark 5.1.

a. (Order of schemes) In general, our discretizations of the non-local term in (1.1)are at most first order accurate. In the case γ ∈ [1, 2), a first order rate is obtainedby choosing ∆z = ∆x1/2. Higher order discretizations can be derived using higherorder quadrature and interpolation rules, but the resulting discretizations are notmonotone in general. On the other hand, if η ≡ z, then interpolation is not neededand consequently the monotone discretizations of Section 5.3 are 2nd order accurate.

b. (Remaining discretizations) To obtain fully discrete schemes it remains todiscretize the various terms involving bαDφ, for example by quadrature and finitedifferencing, cf. Appendix A. In applications, the densities k and k can often beexplicitly calculated, e.g., using incomplete gamma functions as in [1]. Otherwisethese quantities also have to be computed by quadrature. Furthermore, regardingSections 5.2.2 and 5.3.2, it also remains to discretize the surface integral in y. Thisdiscretization does not pose any problems, neither numerically nor in the analysis,as long as positive quadratures are used. The details are left to the reader.


c. (Increasing efficiency) From a practical point of view in terms computationalefficiency, quadratures should be implemented using FFT. This is standard and werefer to, e.g., [22] for the details.

d. (Generalization I) The above approximations (with obvious modifications)also apply to integral terms of the type

Jα[φ](t, x) =M∑i=1

∫R\{0}

(φ(t, x+ ηαi (t, x, z))− φ(t, x)− ηαi (t, x, z)Dφ(t, x)

)ki(z)dz.

Such terms appear in M -dimensional Levy models based on M independent Poissonrandom measures coming from one-dimensional Levy processes. This is a rich classof models with many applications. For more information and analysis of suchmodels we refer to the book [38].

e. (Generalization II) With obvious modifications, our approximations also applyto linear and non-linear equations involving the fractional Laplace operator

(−∆)αu(x) = cα

∫|z|>0

u(x+ z)− u(x)− zDu(x)|z|N+2α

dz, α ∈ (0, 1),

where x, z ∈ RN and cα is a constant, in which case the Levy measure takes theform ν(dz) = |z|−N−2αdz. This measure satisfies (A.3) except for the “exponentialdecay at infinity” requirement. It is straightforward to recast the entire theory toallow for a fractional Laplace setting where assumption (A.3) is replaced by∫

|z|>0

|z|2 ∧ 1 ν(dz) <∞.

6. Error estimates for a switching system approximation

In this section we obtain error estimates for a switching system approximationof (1.1)–(1.2). This result, which has independent interest, plays a crucial role inthe proof of Theorem 4.2 in Section 7.

The switching system will be written as

Fi(t, x, v, ∂tvi, Dvi, D2vi, ui(t, ·)) = 0 in QT , i ∈ {1, 2, .....,m}, (6.1)

v(0, x) = (g(x), . . . , g(x)) in RN , (6.2)

where v = (v1, . . . , vm) is in Rm and for sets Ai such that ∪iAi = A,

Fi(t, x, r, pt, px, X, φ(·))= max

{pt + sup

α∈Ai

[Lα(t, x, ri, px, X)− Jα[φ](t, x)

]; ri −Mir

},

Lα(t, x, r, p,X) := −tr(aα(t, x)X

)− bα(t, x) · p+ cα(t, x)r − fα(t, x),

Mir = minj 6=i{rj + k}, k > 0,

for(t, x, r, pt, px, X

)∈ R × RN × Rm × R × RN × SN and any smooth bounded

function φ. The operator Jα[φ] is defined below (1.1).In the pure PDE case, such approximations have been studied in, e.g., [17, 23, 8].

Here we extend the error estimates of [8] to non-local Bellman equations. In acomplimentary article [12], we develop a viscosity solution theory covering switchingsystems like (6.1). We refer to that paper for the precise definition of viscositysolutions and proofs of the associated results utilized herein. If assumptions (A.1)– (A.4) of Section 3 hold, then we have the following well posedness result [12]:

Proposition 6.1. Assume that conditions (A.1)–(A.4) hold. There exists a uniqueviscosity solution v of (6.1)–(6.2), satisfying |v|1 ≤ C for a constant C depending


only on T and K from (A.1)–(A.3). Furthermore, if w1, w2 are respectively vis-cosity sub and supersolutions of (6.1) satisfying w1(0, ·) ≤ w2(0, ·), then w1 ≤ w2.

Before we continue, we need the following remark.

Remark 6.1. The functions σα, bα, cα, fα, ηα are only defined for times t ∈ [0, T ].But they can be easily extended to times [−r, T + r] for any r > 0 in such a waythat (A.1) – (A.3) still hold. In view of Proposition 6.1 we can then solve theinitial value problem up to time T + r or, by using a translation in time, we maystart from time −r. We will use these facts several times below.

By equi-continuity and the Arzela-Ascoli theorem it easily follows that the eachcomponent of the solution of (6.1)–(6.2) converges locally uniformly to the solutionof (1.1)–(1.2) as k → 0. To derive an error estimate we use Krylov’s method ofshaking the coefficients coupled with an idea of P.-L. Lions as in [8]. We need thefollowing auxiliary system

F εi (t, x, vε, ∂tvεi , Dvεi , D

2vεi , vεi (t, ·)) = 0 in QT+ε2 , i ∈ {1, . . . ,m}, (6.3)

vε(0, x) = (g(x), . . . , g(x)) in RN ,

where vε = (vε1, . . . , vεm) and

F εi (t, x, r, pt, px, X, φ(·)) = max{pt + sup

α∈Ai;|e|≤ε;0≤s≤ε2

(Lα(t+ s, x+ e, ri, px, X)

− Jα(t+ s, x+ e)φ); ri −Mir

}.

The operators L, J , and M are as previously defined.Note that we have used the extension of the data mentioned in Remark 6.1. By

regularity and continuous dependence results from [12] we have

Proposition 6.2. Assume that (A.1)–(A.4) hold. There exists a unique viscositysolution vε : QT+ε2 → R of (6.3) satisfying

|vε|1 +1ε|vε − v|0 ≤ C,

where C depends T and K. Furthermore, if w1 and w2 are respectively sub andsupersolutions of (6.3) satisfying w1(0, ·) ≤ w2(0, ·), then w1 ≤ w2.

We are now in a position to prove the following main result of this section:

Theorem 6.3. Assume that (A.1)–(A.4) hold. If u and v are respectively viscositysolutions of (1.1)–(1.2) and (6.1)–(6.2), then for sufficiently small k,

0 ≤ vi − u ≤ Ck13 , i ∈ {1, . . . ,m},

where C depends only on K and T .

Proof. Since w = (u, . . . , u) is a viscosity subsolution of (6.1), the first inequalityu ≤ vi follows from the comparison principle.

The second inequality will be obtained in the following. Since vε is the viscositysolution of (6.3), it follows that

∂tvεi + sup

α∈Ai

(Lα(t+ s, x+ e, vεi (t, x), Dvεi , D

2vεi )− Jα(t+ s, x+ e)vεi)≤ 0

in QT+ε2 in the viscosity sense, i = 1, . . . ,m. After a change of variable, we concludethat for every 0 ≤ s ≤ ε2 and |e| ≤ ε, vε(t − s, x − e) is a viscosity subsolution ofthe uncoupled system

∂twεi + sup

α∈Ai

(Lα(t, x, wεi , Dw

εi , D

2wεi )− Jα(t, x)wεi)

= 0 in QεT , (6.4)

where QεT := (ε2, T )× RN . Now set vε := vε ? ρε, where ρε is the mollifier definedin (1.3). A Riemann-sum approximation shows that this function is the limit of


convex combinations of viscosity subsolutions v(t − s, x − e) of the convex system(6.4). Hence vε is also a viscosity subsolution of (6.4) (see the appendix of [29] formore details). On the other hand, since vε is a continuous subsolution of (6.3),

vεi ≤ minj 6=i

vεj + k in QT+ε2 , i ∈ {1, . . . ,m}.

It follows that maxi vεi (t, x)−min vεj(t, x) ≤ k in QT+ε2 , and therefore

|vεi − vεj |0 ≤ k, i, j ∈ {1, . . . ,m}.Then, by the definition and properties of vε,

|∂tvεi − ∂tvεj |0 ≤ Ck

ε2and |Dnvεi −Dnvεj |0 ≤ C

k

εn,

for n ∈ N, i, j ∈ {1, . . . ,m}, where C depends only on ρ, T , and K. For ε < 1, itfollows that

|∂tvεj + supα∈Ai

(Lα(t, x, vεj(t, x), Dvεj , D2vεj)− Jα(t, x)vεj

)− ∂tvεi − sup

α∈Ai

(Lα(t, x, vεi(t, x), Dvεi, D2vεi)− Jα(t, x)vεi

)| ≤ C k

ε2

and, since vε is subsolution of (6.4),

∂tvεi + supα∈A

(Lα(t, x, vεi(t, x), Dvεi, D2vεi)− Jα(t, x)vεi

)≤ C k

ε2in QεT ,

where the constant C depends on ρ, T , and K. From this inequality it is easy tosee that vεi − teKtC k

ε2 is a subsolution of (1.1) restricted to QεT . Hence, by thecomparison principle,

vεi − u ≤ eKt(|vεi(ε2, ·)− u(ε2, ·)|0 + Ct

k

ε2

)in QεT , i ∈ {1, . . . ,m}.

By regularity in time, |u(t, ·) − vi(t, ·)|0 ≤ (|u|1 + |vi|1)ε, and by Proposition 6.2and properties of mollification we conclude that

vi − u ≤ vi − vεi + vεi − u ≤ C(ε+k

ε2) in QT , i ∈ {1, . . . ,m}.

Now the theorem follows by minimizing with respect to ε. �

7. The Proof of Theorem 4.2

To prove Theorem 4.2 we will use different arguments for the upper and lowerbounds. The upper bound, part (a), is the “easy” part, and it is essentially areformulation of the general upper bound established in [29]. We skip the details,and prove only part (b) which is a new result.

Proof of Theorem 4.2 (b). Without loss of generality we will assume thatA is finite:

A = {α1, α2, . . . , αm}.The proof of this statement is similar to the one given in [8] in the pure PDE caseand relies on assumption (A.1). Now we follow [8] and use a switching systemapproximation to construct approximate supersolutions of (1.1) which are point-wise minima of smooth functions and approximates the viscosity solution of (1.1)–(1.2). Consider

F εi(t, x, vε, ∂tv

εi , Dv

εi , D

2vεi , vεi (t, ·)

)= 0 in QT+2ε2 (7.1)

vε(0, x) = v0(x) in RN ,

where vε = (vε1, . . . , vεm), v0 = (g, . . . , g), and

F εi(t, x, r, pt, px, X, φ(t, ·)

)


= max{pt + min

0≤s≤ε2,|e|≤ε

(Lαi(t+ s− ε2, x+ e, r, pt, px, X

)− Jαi(t+ s− ε2, x+ e)φ

); ri −Mir

},

where L, J and M are defined below (6.1) in Section 6. This new problem is well-posed and each component of the solution of this switching system will converge tothe viscosity solution of (1.1) as k, ε→ 0:

Lemma 7.1. Assume that conditions (A.1)–(A.4) hold. There exists a uniquesolution vε of (7.1) satisfying

|vε|1 ≤ K, maxi,j|vεi − vεj |0 ≤ k, and for k small, max

i|u− vεi | ≤ C(ε+ k

13 ),

where K, C only depend on T and K from (A.2)–(A.4).

Proof. From [12] we have the existence and uniqueness of a viscosity solution, andmoreover the uniform bounds

|vε|1 ≤ K and |vε − v0|0 ≤ Cε,

where v0 is the unique viscosity solution of (7.1) corresponding to ε = 0. Thelast inequality in the lemma now follows since |u − v0

i |0 ≤ Ck13 by Theorem 6.3.

To second inequality follows since arguing as in the proof of Theorem 6.3 leads to0 ≤ maxi vεi −minj vεj ≤ k in QT+2ε2 . �

Next we time-shift and mollify vε. For i = 1, . . . ,m, set

vεi (t, x) := vεi (t+ ε2, x), vεi(t, x) := ρε ? vεi (t, x),

where ρ is defined in (1.3). Note that supp(ρε) ⊂ (0, ε2) × B(0, ε) and that thefunctions vε, vε, vε are well-defined respectively on QT+2ε2 , (−ε2, T + ε2] × RN ,QT+ε2 . By Lemma 7.1 and properties of mollifiers,

|vε|1 ≤ K, |vε − vε|0 ≤ Kε,maxi,j|vεi − vεj | ≤ C(k + ε) in QT+ε2 , (7.2)

maxi|u− vε,i| ≤ C(ε+ k

13 ) in QT ,

where C depends only on ρ and K,T from (A.2)–(A.4). A supersolution of (1.1)can now be produced by setting

w := minivεi.

Lemma 7.2. Assume that conditions (A.1)—(A.4) hold and ε ≤ (8 supi[vεi ]1)−1k.For every (t, x) ∈ QT , if j := argminivεi(t, x),

∂tvεj + Lαj(t, x, vεj(t, x), Dvεj(t, x), D2vεj(t, x)

)− Jαj (t, x)vεj ≥ 0. (7.3)

We postpone the proof of this lemma. From this lemma it follows that w is anapproximate supersolution to the scheme (4.1) when ε ≤ (8 supi[vεi ]1)−1k:

S(h, t, x, w(t, x), [w]t,x

)≥ −E2(K, h, ε) in G+

h , (7.4)

where K comes from Lemma 7.1. To see this, let (t, x) ∈ QT and set j :=argminivεi(t, x). At (t, x), w(t, x) = vεj(t, x) and w ≤ vεj in Gh. Hence (S1)implies that

S(h, t, x, w(t, x), [w]t,x

)≥ S

(h, t, x, vεj(t, x), [vεj ]t,x

).

By consistency (S3)(ii) we have

S(h, t, x, vεj(t, x), [vεj ]t,x

)≥ ∂tvεj + F

(t, x, vεj(t, x), Dvεj , D2vεj , vεj(t, ·)

)− E2(K, h, ε),


≥ ∂tvεj + Lαj(t, x, vεj(t, x), Dvεj(t, x), D2vεj(t, x)

)− Jαj (t, x)vεj − E2(K, h, ε),

and (7.4) then follows from Lemma 7.2.To derive the lower bound on the error uh − u, we take ε = (8 supi[vεi ]1)−1k and

use (7.4) and comparison Lemma 4.1 to get

uh − w ≤ eµt|(gh − w(0, ·))+|+ 2teµtE2(K, h, ε) in Gh.

By (7.2), |w − u| ≤ C(ε+ k + k13 ), and hence

uh − u ≤ eµt|(gh − w(0, ·))+|+ 2teµtE2(K, h, ε) + C(ε+ k + k13 ) in Gh,

possibly with a new constant C. Since ε = Ck, the proof is complete by minimizingthe right hand side with respect to ε. �

Proof of Lemma 7.2. We begin by fixing (t, x) ∈ QT and set j = argmini vεi(t, x).Then

vεj(t, x)−Mjvε(t, x) = maxi 6=j

{vεj(t, x)− vεi − k

}≤ −k.

Therefore, by the Holder continuity of vε and basic properties of mollifiers,

vεj(t, x)−Mj vε(t, x) ≤ −k + 2 maxi

[vεi ]12ε

and

vεj(s, y)−Mj vε(s, y) ≤ −k + 2 maxi

[vεi ]1(2ε+ |x− y|+ |t− s| 12 ),

for all (s, y) ∈ QT . Consequently, if |x−y| < ε, |t−s| < ε2, and ε ≤ (8 supi[vεi ]1)−1k,then

vεj(s, y)−Mj vε(s, y) < 0. (7.5)

To continue we need the following remark. Let u1, . . . , uk be functions satisfy-ing (7.5) at (t, x), then any linear combination uλ =

∑ki=1 λiu

i with λi ≥ 0 and∑ki=1 λi = 1, also satisfies (7.5) at (t, x). If, in addition, u1, . . . , uk are supersolu-

tions of

max{∂tuj + Lαj (t, x, uj , Duj , D2uj)− Jαj [uj ](t, x);uj −Mju

}= 0, (7.6)

then in view of (7.5) they are also supersolutions of the linear equation

∂tuj + Lαj (t, x, uj , Duj , D2uj)− Jαj [uj ](t, x) = 0 (7.7)

at (t, x). An easy adaptation of the proof of Lemma 6.3 in [29] then shows that uλ

is also a viscosity supersolution of (7.7) at (t, x).A change of variables reveals that

{vε(· − s, · − e)

}s,e

, 0 ≤ s < ε2, |e| < ε, isa family of supersolutions of (7.6) with [vεj −Mj vε](t − s, x − e) < 0. We notethat by approximating the function vεj by a Riemann sum, we see that it is thelimit of convex combinations of vε(· − s, · − e). In view of the above remark, theseconvex combinations are supersolutions of (7.6), and hence by the stability resultfor viscosity supersolutions, so is the limit vεj . Finally, since this function is smoothit is also a classical supersolution of (7.6) and hence (7.3) holds. �


Appendix A. An example of a monotone discretization of Lα

Let {ei}Ni=1 be the standard basis of RN and aij the ij-th element of the matrixa. Kushner and Dupuis [33] suggest the following discretization of Lα in (1.1):

Lαhφ :=N∑i=1

[aαii∆ii +

∑i 6=j

(aα+ij ∆+

ij − aα−ij ∆−ij

)+ bα+

i δ+i − b

α−i δ−i

]φ,

where b+ = max{b, 0}, b− = (−b)+, and

δ±i φ(x) = ± 1∆x{φ(x± ei∆x)− φ(x)

},

∆iiφ(x) =1

∆x2

{φ(x+ ei∆x)− 2φ(x) + φ(x− ei∆x)

},

∆+ijφ(x) =

12∆x2

{2φ(x) + φ(x+ ei∆x+ ej∆x) + φ(x− ei∆x− ej∆x)

}− 1

2∆x2

{φ(x+ ei∆x) + φ(x− ei∆x) + φ(x+ ej∆x) + φ(x− ej∆x)

}∆−ijφ(x) =

12∆x2

{2φ(x) + φ(x+ ei∆x− ej∆x) + φ(x− ei∆x+ ej∆x)

}− 1

2∆x2

{φ(x+ ei∆x) + φ(x− ei∆x) + φ(x+ ej∆x) + φ(x− ej∆x)

}.

By Taylor expansion it is easy to check that the truncation error is given by (3.3).Moreover Lh can be written in the form (3.5) with

lα,nh,β,β±ei =1

∆x2

[aαii(t, x)− 1

2

∑j 6=i

|aαij(t, x)|]

+bα±i (t, x)

∆x,

lα,nh,β,β+ei±ej =aα±ii (t, x)

∆x2, lα,nh,β,β−ei±ej =

aα±ii (t, x)∆x2

, i 6= j,

and lα,nh,β,β

= 0 otherwise. This approximation is monotone if lα,nh,β,β

≥ 0 for allα, β, β, n and ∆x > 0, which happens to be the case, e.g., if a is diagonally domi-nant:

aαii(t, x)−∑j 6=i

|aαij(t, x)| ≥ 0 in QT , for each α ∈ A.

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(Imran H. Biswas)Seminar for Applied Mathematics, ETH, CH-8092 Zurich, Switzerland

E-mail address: [email protected]

(Espen R. Jakobsen)

Norwegian University of Science and Technology, NO–7491, Trondheim, Norway


(Kenneth Hvistendahl Karlsen)

Centre of Mathematics for Applications, University of Oslo, P.O. Box 1053, Blindern,NO–0316 Oslo, Norway


URL: folk.uio.no/kennethk

Difference-Quadrature Schemes for Nonlinear Degenerate Parabolic Integro-PDE

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